On Specht modules of general linear groups. (Q1414687)

Let \(F\) be a field of characteristic \(p\) where \(p\nmid q\) and let \(S^\lambda\) be a Specht module for \(\text{GL}(\mathbb{F}_q)\) over \(F\) as in [\textit{G. D. James}, Representations of general linear groups, Lond. Math. Soc. Lect. Note Ser. 94 (1984; Zbl 0541.20025)]. A nonzero vector in \(S^\lambda\) is a linear combination of \(\lambda\)-flags, but not every linear combination occurs. The author shows in particular that if \(T\) is the row standard tableaux defined by a term in the linear combination that is maximal according to a certain partial order, then \(T\) must be a standard tableaux. This is analogous to a result for Specht modules of the symmetric group, but in contrast to that case one does not have a simple connection between standard tableaux and elements of some basis of \(S^\lambda\).